= Solution
Two examples are <sodium dodecyl sulfate>, with a negatively charged sulfate head and a hydrocarbon tail, and <cetrimonium bromide>, with a positively charged ammonium head and a hydrocarbon tail. Both are <amphiphiles>: one region is <hydrophilic> and another is <hydrophobic>.
In <water>, the <hydrophobic effect> favors association of the tails into a core, reducing their exposed area while the hydrated heads remain outside. This can lower the total <Gibbs free energy> despite loss of translational <entropy>. Head-group repulsion, interfacial cost and chain-packing constraints oppose unlimited growth, so finite <micelles> are favored under suitable conditions above the <critical micelle concentration>.
The appropriate geometric measure is the <amphiphile packing parameter>:
$$
P=\frac{v}{a_0\ell}=\frac{0.12}{0.5\times0.9}=\boxed{0.267}.
$$
For a spherical <micelle> with $n$ tails and core radius $r_c$, the volume and area relations are $nv=4\pi r_c^3/3$ and $na_0=4\pi r_c^2$. Dividing gives $r_c=3v/a_0$, so a sphere fits without overextending a tail if $P\leq1/3$. Here $P<1/3$ and
$$
r_c=\frac{3v}{a_0}=0.72\,\mathrm{nm}<\ell.
$$
Thus the packing model predicts \b[spherical <micelles>]. This is a preferred aggregate geometry, not a claim that concentration, temperature or ionic conditions can never alter the morphology.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2019-scm-micelle-packing.png]
{title=An original cross-section of spherical micelle packing}
{description=Hydrophilic heads face the water and hydrophobic tails fill the core; the drawing illustrates packing geometry rather than a measured aggregation number.}
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